Why am I not able to correctly calculate the median survival time for the Weibull distribution?

Question

The survival function for the Weibull distribution is:

$S(t) = exp(-\lambda t^\alpha)$, $\lambda$ = scale parameter, $\alpha$ = shape parameter

If I wanted to calculate the median survival time, I would set $S(t) = 0.5$. Rearranging terms, this means the median survival time would be calculated as follows:

$t_{median} = (\frac{-log(0.5)}{\lambda})^{(1/\alpha)}$

Let's say I set $\lambda = 3$ and $\alpha = 2$. Then the median survival time is clearly:

$t_{median} = (\frac{-log(0.5)}{3})^{(1/2)} = 0.481$

However, when I do a sanity check on this and run this in R, the median survival for a Weibull distribution with scale = 3 and shape = 2 is clearly not 0.481 and is more like 2.5:

Time <- sort(rweibull(1000, shape = 2, scale = 3))
Time[500]

Any time I run this code, I am getting somewhere right around 2.5, +/- 0.1 or so. But definitely nowhere near 0.481.

So what do I have wrong here? Why is the actual median time for simulated data nowhere near the theoretical median time?

Answer 1

You got trapped in the thicket of multiple parameterizations of the Weibull distribution.

In what Wikipedia calls the "standard parameterization", the survival function with scale parameter $\lambda$ and shape parameter $\alpha$ is:

$$S(t)=\exp(-(t/\lambda)^\alpha). $$

That's the parameterization used by rweibull(). In that parameterization, the median is $\lambda(\ln 2)^{(1/\alpha)}$, which for $\lambda= 3$ and $\alpha = 2$ gives:

3*(log(2)^(1/2))
# [1] 2.497664

presumably close to the value that you found by simulation.

You seem to be using what Wikipedia calls the "first alternative parameterization."